About Solving Exponential and Logarithmic Inequalities:
When working with exponential or logarithmic inequalities, we have different options based on the given problem. For easier exponential inequalities, we can set up an inequality using the exponents. For harder problems featuring both exponential and logarithmic inequalities, we turn to the test-point approach.
Test Objectives
- Demonstrate the ability to solve an exponential inequality
- Demonstrate the ability to solve a logarithmic inequality
#1:
Instructions: Solve each inequality.
$$a)\hspace{.2em}4^{2x}\cdot 64^{x + 1}> \left(\frac{1}{16}\right)^{x + 2}$$
$$b)\hspace{.2em}\left(\frac{1}{9}\right)^{2x}\cdot 81^{-4x}≤ \left(\frac{1}{3}\right)^{-4}$$
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#2:
Instructions: Solve each inequality.
$$a)\hspace{.2em}\left(\frac{1}{4}\right)^{x}\cdot \left(\frac{1}{4}\right)^{2x + 3}< 64$$
$$b)\hspace{.2em}{-}2e^{5x + 5}+ 3 > -47$$
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#3:
Instructions: Solve each inequality.
$$a)\hspace{.2em}\left(\frac{1}{5}\right)^{x}+ 25^{-x}≥ 30$$
$$b)\hspace{.2em}(5x + 2)^{x^2 - x - 2}≤ 1$$ $$\text{and}$$ $$5x + 2 > 0$$
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#4:
Instructions: Solve each inequality.
$$a)\hspace{.2em}9\log_6(5x - 9) ≥ 27$$
$$b)\hspace{.2em}\log_5(x - 3) + \log_5(x + 1) ≤ 1$$
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#5:
Instructions: Solve each inequality.
$$a)\hspace{.2em}\log^{2}(x) + \log(x) < 2$$ Note: log2(x)=(log(x))2
$$b)\hspace{.2em}\log_3(x) - 2\log_{\frac{1}{3}}(x) > 6$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}x > -1$$ Interval Notation: $$(-1, \infty)$$ Graphing the Interval: 
$$b)\hspace{.2em}x ≥ -\frac{1}{5}$$ Interval Notation: $$\left[-\frac{1}{5}, \infty\right)$$ Graphing the Interval: 
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#2:
Solutions:
$$a)\hspace{.2em}x > -2$$ Interval Notation: $$\left(-2, \infty\right)$$ Graphing the Interval: 
$$b)\hspace{.2em}x < \frac{\ln(25) - 5}{5}$$ Interval Notation: $$\left(-\infty, \frac{\ln(25) - 5}{5}\right)$$ Graphing the Interval: 
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#3:
Solutions:
$$a)\hspace{.2em}x ≤ -1$$ Interval Notation: $$\left(-\infty, -1\right]$$ Graphing the Interval: 
$$b)\hspace{.2em}{-}\frac{1}{5}≤ x ≤ 2$$ Interval Notation: $$\left[-\frac{1}{5}, 2\right]$$ Graphing the Interval: 
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#4:
Solutions:
$$a)\hspace{.2em}x ≥ 45$$ Interval Notation: $$\left[45, \infty\right)$$ Graphing the Interval: 
$$b)\hspace{.2em}3 < x ≤ 4$$ Interval Notation: $$\left(3, 4\right]$$ Graphing the Interval: 
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#5:
Solutions:
$$a)\hspace{.2em}\frac{1}{100}< x < 10$$ Interval Notation: $$\left(\frac{1}{100}, 10\right)$$ Graphing the Interval: 
$$b)\hspace{.2em}x > 9$$ Interval Notation: $$\left(9, \infty\right)$$ Graphing the Interval: 
